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Turbulent boundary-layer flow beneath a vortex. Part 2. Power-law swirl

Published online by Cambridge University Press:  03 April 2020

David E. Loper Open the ORCID record for David E. Loper [Opens in a new window]
David E. Loper*
Affiliation:
Professor Emeritus, Florida State University, Tallahassee, FL32306, USA
*
†Email address for correspondence: [email protected]
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Abstract

The problem formulated in Part 1 (Loper, J. Fluid Mech., vol. 892, 2020, A16) for flow in the turbulent boundary layer beneath a vortex is solved for a power-law swirl: $v_{\infty }(r)\sim r^{2\unicode[STIX]{x1D703}-1}$, where $r$ is cylindrical radius and $\unicode[STIX]{x1D703}$ is a constant parameter, with turbulent diffusivity parameterized as $\unicode[STIX]{x1D708}=v_{\infty }L$ and the diffusivity function $L$ either independent of axial distance $z$ from a stationary plane (model A) or constant within a rough layer of thickness $z_{0}$ adjoining the plane and linear in $z$ outside (model B). Model A is not a useful model of vortical flow, whereas model B produces realistic results. As found in Part 1 for $\unicode[STIX]{x1D703}=1.0$, radial flow consists of a sequence jets having thicknesses that vary nearly linearly with $r$. A novel structural feature is the turning point $(r_{t},z_{t})$, where the primary jet has a minimum height. The radius $r_{t}$ is a proxy for the eye radius of a vortex and $z_{t}$ is a proxy for the size of the corner region. As $r$ decreases from $r_{t}$, the primary jet thickens, axial outflow from the layer increases and axial oscillations become larger, presaging a breakdown of the boundary layer. For small $\unicode[STIX]{x1D703}$, $r_{t}\sim z_{0}/\unicode[STIX]{x1D716}\unicode[STIX]{x1D703}$ and $z_{t}\sim z_{0}/\unicode[STIX]{x1D703}^{3/2}$. The lack of existence of the turning point for $\unicode[STIX]{x1D703}\gtrsim 0.42$ and the acceleration of the turning point away from the origin of the meridional plane as $\unicode[STIX]{x1D703}\rightarrow 0$ provide partial explanations why weakly swirling flows do not have eyes, why strongly swirling flows have eyes and why a boundary layer cannot exist beneath a potential vortex.

JFM classification

Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Atmospheric flows Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A17
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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